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Undid revision 1217006437 by SovietWizard (talk). It should be +1/2 not -1/2, as otherwise it is the inverse factorial function, not inverse gamma. |
Undid revision 1306141408 by 2601:1C0:5780:8D40:D04C:4026:7A75:7E14 (talk) Gamma(5) is 4 factorial and is definitely 24 |
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{{Short description|Inverse of the gamma function}}
{{Distinguish|Inverse-gamma distribution|Reciprocal gamma function}}
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To compute the branches of the inverse gamma function one can first compute the [[Taylor series]] of <math>\Gamma(x)</math> near <math>\alpha</math>. The series can then be truncated and inverted, which yields successively better approximations to <math>\Gamma^{-1}(x)</math>. For instance, we have the quadratic approximation:<ref>{{cite conference |first1=Robert M.|last1=Corless |first2=Folitse Komla|last2=Amenyou |last3=Jeffrey |first3=David |book-title=2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC) |title=Properties and Computation of the Functional Inverse of Gamma |conference=International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC) |date=2017 |pages=65 |doi=10.1109/SYNASC.2017.00020|isbn=978-1-5386-2626-9 |s2cid=53287687 }}</ref>
<math display="block"> \Gamma^{-1}\left(x\right)\approx\alpha+\sqrt{\frac{2\left(x-\Gamma\left(\alpha\right)\right)}{\
where <math> \psi^{\left(1 \right)} \left(x \right)</math> is the [[trigamma function]]. The inverse gamma function also has the following [[asymptotic formula]]<ref>{{cite thesis |type=MS |last1=Amenyou |first1=Folitse Komla |last2=Jeffrey |first2=David |title="Properties and Computation of the inverse of the Gamma Function" |date=2018 |pages=28 |url=https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=7340&context=etd}}</ref>
<math display="block"> \Gamma^{-1}(x)\sim\frac{1}{2}+\frac{\ln\left(\frac{x}{\sqrt{2\pi}}\right)}{W_{0}\left(e^{-1}\ln\left(\frac{x}{\sqrt{2\pi}}\right)\right)}\,,</math>
where <math>W_0(x)</math> is the [[Lambert W function]]. The formula is found by inverting the [[Stirling's approximation|Stirling approximation]], and so can also be expanded into an asymptotic series.
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== References ==
{{reflist}}
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[[Category:Gamma and related functions]]
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